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We need to find whether the vector AC is a multiple of the vector AB
__
AC = < 4-6, -5-7, 1-1 > = < -2, -12, 0 > 

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AB = < 2-6, -3-7, 1-1 > = < -4, -10, 0 >

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AC is not a multiple of AB, so A,B, and C are not collinear.

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You can also do it by finding the distances between the three points,
and determining whether the sum of two of the distances AB, BC, AC are
equal to the third distance:

AB =  =  =  =  =  = 

AC =  =  =  =  =  = 

BC =  =  =  =  =  =  = 

No two of those have the sum of the smallest two,  +  
does not equal the largest , so the three points cannot be
collinear. 
 

Edwin